Comprehensive investigation of intermittent large-amplitude excursions in the memristive Hindmarsh-Rose neuron model

Abstract This study systematically investigates intermittent large-amplitude oscillations across all three state variables of the memristive Hindmarsh–Rose neuron model. The large-amplitude excursions, arising from interior crisis-induced intermittency, are first identified through local and two-parameter bifurcation analysis. All three state variables are then statistically characterized using the significant-height threshold criterion, probability distribution functions, probability of exceedance, and the density of threshold exceeding peaks over successive time-windows. The results reveal that the membrane potential exhibits a consistently lower probability and density of extreme events compared to the recovery and slow variables, and that these extreme events are confined to the chaotic regions identified via two-parameter Lyapunov exponent maps. To examine the robustness of this behavior under memory effects, the analysis is extended to the fractional-order memristive Hindmarsh-Rose neuron model, where the period-doubling route to chaos and the corresponding exceedance statistics of all state variables are investigated. The membrane potential retains its comparatively lower susceptibility to extreme events across fractional orders, confirming that this variable-dependent behavior is a robust feature of the neuron model rather than an artifact of the integer-order formulation.